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STAT 100A HWI
Note
(1) If all the outcomes are equally likely, then Pr(
A
) =

A

/

Ω

.
(2) Conditional probability Pr(
A

B
) = Pr(
A
∩
B
)
/
Pr(
B
).
Please show all the necessary steps in your calculations. Please be precise with notation.
Problem 1:
Suppose we ﬂip a fair coin 4 times independently.
(1) What is the sample space?
(2) What is the set that corresponds to the event that the number of heads is 2? What is its
probability?
(3) Let
Z
i
= 1 if the
i
th ﬂip is head, and
Z
i
= 0 otherwise, for
i
= 1
,
2
,
3
,
4. Let
X
be the
number of heads. Express
X
in terms of
Z
i
.
(4) What is the probability distribution of
X
? That is, what is Pr(
X
=
k
) for
k
= 0
,
1
,
2
,
3
,
4?
Problem 2:
Suppose we roll a fair die twice independently. Let
X
and
Y
be the two numbers we
get.
(1) What is the sample space? Let
A
be the event that
X >
4, and
B
be the event that
Y >
4.
What are Pr(
A
), Pr(
B
)?
(2) Let
C
be the event that min(
X,Y
)
>
4? What is Pr(
C
)? What is the relationship between
Pr(
C
) and Pr(
A
), Pr(
B
)?
(3) Let
D
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